Anumber 10 awg copper wire has a resistance of 0.9989 ohms per 1,000 feet and a cross sectional area of 10,380 cm. if the resistance of copper wire is inversely proportional to its cross-sectional area, what would be the resistance of 1,000 feet of copper wire with a cross section of 5,178 cm? (round the final answer to four decimal places.)

The resistance of a wire is calculated by the following formula:

    (1)

Where:

is the resistivity of the material the wire is made of. In this case we are talking about copper, and its resistivity at is:

is the length of the wire, which in this case is , but we will make the conversion to meters, in order to work with the same units:

is the transversal area of the wire. In this case is a circumference, so we will use the formula of the area of the circumference:

Now, in this problem we have two transversal areas:

And two resistances:

and   which is the one we are asked to find.    

Taking into account we are working with the same material (copper), the will be the same. In addition, the length of the wire is the same, the only thing that changes is the transversal area.

According to this given data, we have a system of two equations, as follows:

    (2)

    (3)

We know all the values, except .

At this point of the problem, we will approach it in the following way:

From (2) let’s isolate :

    (4)

From (3) let’s also isolate :

    (5)

If equation (4)=equation (5)

   (6)  

Now, we have to find the value of :

   (7)  

Finally:

Rounding:

>>>>>This is the result

It is important to note that as the resistance of the wire is inversely proportional to its cross-sectional area, when the area decreases the resistance increases.